Enhancing second harmonic generation by Q-boosting lossless cavities beyond the time bandwidth limit

2.1 Coupled mode theory for doubly-resonant second harmonic cavities

Here, we employ CMT to describe SHG in a doubly-resonant cavity, which supports modes at both FF, ω ₁, and SH frequency, ω ₂ = 2ω ₁, with amplitudes a ₁ and a ₂, respectively [41]. The cavity is excited by an external source s 1 + , at frequency ω _p = ω ₁ (resonant excitation). In general, the set of equations describing SHG in a doubly-resonant cavity is [40, 42]:

where Q ₁ and Q ₂ are the Q-factors of modes 1 (FF) and 2 (SH), β ₁ and β ₂ are the internal cavity coupling coefficients, which are responsible for pump depletion and SH conversion, respectively, and they are related to the overlap integral between the FF and SH modes [40, 43]. As pointed out in Ref. [40], we set β ₂ = β ₁/2 to fulfill the conservation energy constraint. The terms s 1 − and s 2 − represent the outgoing waves at ω ₁ and ω ₂, respectively. Although not being explicitly reported, time dependence of a _k and s k ± terms in Equation (1) is implied. In analogy with [40], we define the SH (FF) power conversion efficiency as ζ S H = P 2 − / P 1 + ζ F F = P 1 − / P 1 + , where P 2 − P 1 − is the output power at ω ₂ (ω ₁) and P 1 + is the input power at ω _p = ω ₁. First, we focus on the SH power conversion within a cavity where Q ₁ is constant over time (see Figure 1a). When the input source is a monochromatic continuous-plane-wave (CW), complete power conversion efficiency ( ζ S H = s 2 , s s − 2 / s 1 + 2 = 1 [40]) can be achieved for an optimum value of Q ₁ Q 1 opt in the steady state (ss) condition [40]. As an example, we calculate ζ _SH for various values of Q ₁ by solving Equation (1) and we display the results as a function of the normalized Q-factor θ 1 = Q 1 / Q 1 opt in Figure 1a (red curve) for the following set of parameters: ω ₁ = 0.791 rad/fs (ν = 125.9 THz), β ₁ = 1.5 × 10⁻⁴ 1 /J , Q ₂ = 1500, and s 1 + = 17.33 W . The value of Q 1 opt is bounded to ω ₁, β ₁, Q ₂ and | s 1 + | 2 [40] (see Section 1.1 of the Supplementary Material for more details).

Figure 1:

Framework and design of Q-boosting approach. (a) Static case second harmonic power conversion efficiency (ζ _SH ) as a function of the normalized Q-factor θ 1 = Q 1 / Q 1 opt for a continuous-wave (CW, red curve) and pulsed (black curve, factor × 100 magnification) input sources s 1 + . For both CW and pulsed excitation cases, an optimal set of the cavity parameters exists (see Section 1 of the Supplementary Material). For both cases, quality factor of mode 2 (Q ₂) is 1500. The value of internal coupling (β ₁) is 1.5 × 10⁻⁴ 1 /J for the CW case and 1.5 × 10⁻⁴ fs/J for the pulsed case. (b) Amount of spectral power (shaded region) efficiently stored in a cavity mode (green line) for a narrow (red, top panel) and broadband pulse (black, bottom panel). (c) Time-dependent amplitude modulation (Equation (3)) of the Q-factor at ω ₁ (Q ₁) to enhance second harmonic conversion efficiency. The s 1 + pulse resonantly excites the cavity mode at ω ₁ (bottom panel). Given the coupling (β) between the cavity modes, by carefully tuning the temporal delay (τ) between the arrival of the pulse s 1 + and the time at which the Q ₁ modulation occurs, the relative amplitude of the outgoing waves at ω ₁ s 1 − and ω ₂ = 2ω ₁ s 2 − can be controlled.

Now, we consider a pulsed input source at resonance (ω _p = ω ₁) with Gaussian temporal profile of the form:

where τ _p is the full width at half maximum (FWHM) of the time-dependent intensity profile I t ∝ s 1 + t 2 and s ₀ is the amplitude. s 1 + t and s k − t are the time-dependent amplitudes of the input and output waves, respectively [41]. In analogy with [34, 35], the quantity U k = ∫ − ∞ + ∞ a k t 2 d t defines the time-integrated energy (units [J s]) inside the cavity ascribed to the mode k and the pulse energy of the incoming and outgoing radiation (units [J]) are defined as V 1 + = ∫ − ∞ + ∞ s 1 + t 2 d t = s 0 2 and V k − = ∫ − ∞ + ∞ s k − t 2 d t , respectively. Consequently, the input and output pulse power is calculated as the ratio between the pulse energy and its temporal duration: P k ± = V k ± / Δ t k ± (see Supplementary Material for more details). For a pulsed source, the maximum achievable ζ _SH ζ S H max is much smaller than in the CW case. As shown by the black curve in Figure 1a, for τ _p = 100 fs, β ₁ = 1.5 × 10⁻⁴ fs/J and s ₀ = 17.33 J , ζ S H max ≃ 0.44 % (see Supplementary Material). The difference with the monochromatic CW can be rationalized by recalling that, for a pulsed excitation, the spectral intensity profile (S(ω)) has a bandwidth Δω _p , which is related to the transform-limited pulse duration τ _p by the time-bandwidth product: τ _p ⋅ Δω _p = 4ln2. Therefore, only the fraction of S ω matching the bandwidth of the cavity mode (shaded region in Figure 1b) is involved in the SH conversion process. A Q-boosting approach allow to increase the energy stored inside the cavity by a dynamic control of the Q-factor.

2.2 Time dependent Q-factor modulation

In Figure 1c, we depict the working principle of Q-boosting to enhance SHG. We consider a time-dependent amplitude modulation of Q ₁ (occurring at t = 0), which, in the following part of the work, takes the form

where σ is the switching time, Q _1L and Q _1H being the initial and final values of the FF Q-factor, respectively. In our model, the variation of Q corresponds to the change in the mode bandwidth γ, i.e., Q ₁(t) = ω ₁/(2γ ₁(t)). Moreover, as sketched in Figure 1c, we allow the input pulse s 1 + to enter the cavity with a delay time τ with respect to the instant at which Q ₁ increases. Therefore, from Equation (2), the expression for s 1 + becomes s 1 + t ; τ = s 1 + t − τ . Here, the value of Q ₂ is constant over time and, in the following, we assume Q ₂ = 1500. In general, β ₁ should change as the electric field distribution inside the cavity varies. Here, we assume those modifications to be negligible and consider β ₁ to be constant over time.

First, we numerically solve Equation (1), endowed by the time-dependent Q ₁ as in Equation (3) and assuming Q _1L = 55 and σ = 50 fs. We introduce the modulation amplitude as ρ = Q _1H /Q _1L . The numerical results obtained for ρ = 20 are displayed in Figure 2a, which shows the temporal dynamics of the modes amplitude a 1 t 2 and a 2 t 2 (red and blue curve, respectively), excited by an input pulse with duration τ _p = 100 fs and entering the cavity at τ = −40 fs. The value of |a _k (t)|² (k = 1, 2) is related to the instantaneous energy accumulated in the mode k [41] and decreases exponentially over time due to the ω _k /(2Q _k )a _k term in Equation (1). Here, we introduce the SH energy conversion efficiency as η S H = V 2 − / V 1 + . At this stage it is important to underline that η _SH is related to ζ _SH through the term τ p / Δ t 2 − and the two frameworks allow to extract the same information (see Section 3 of the Supplementary Material). Therefore, we prefer to present the results in terms of η _SH . As displayed in the inset in Figure 2a, the optimum delay time (τ _opt), which maximizes the SH conversion efficiency, is τ _opt ≈ −40 fs (blue circle) and it is of the order of the resonator lifetime Q _1L /ω ₁ ≈ 70 fs, consistent with [35]. Moreover, the inset in Figure 2a shows that, for τ ≪ −τ _p , i.e., the cavity has not switched yet, η _SH is the same as in the static case (SC, η _SH ∼ 7 %) when Q ₁ = Q _1L (dashed purple line in Figure 2a, inset). On the other hand, for τ ≫ τ _p , the input pulse enters the cavity when the switching has already occurred, thus the pulse essentially couples to a high-Q cavity, resulting in low η _SH (since a large portion of the incoming spectrum falls outside the cavity acceptance bandwidth). When −τ _p ≲ τ ≲ τ _p , a large η _SH enhancement is achieved, since the radiation channel at the fundamental frequency reduces after a large portion of the incoming energy pulse has coupled to the cavity. In Figure 2a, η _SH increases from ∼ 7  % to ∼ 67  %. As expected, η F F = V 1 − / V 1 + decreases when η _SH increases.

Figure 2:

Coupled mode theory results within Q-boosted doubly resonant cavities. (a) Squared modulus of the fundamental frequency (FF) mode |a ₁|² (red curve) and squared modulus of the second harmonic (SH) mode |a ₂|² (blue curve) as a function of time t for τ = −40 fs. The inset reports the FF (red curve) and SH (blue curve) conversion efficiency (η _FF and η _SH , respectively) as a function of the temporal delay τ. The blue circle highlights the optimum delay τ ^opt ≈ − 40 fs case. SC: static case. (b) τ ^opt as a function of the initial quality-factor Q _1L for different ratios ρ = Q _1H /Q _1L , where Q _1H is the final quality-factor. The inset reports τ ^opt as a function of ρ for Q _1L = 20.

Now, we discuss how τ _opt relates to ρ. In Figure 2b, we report τ ^opt as a function of Q _1L for different Q-factor ratios ρ = Q _1H /Q _1L with τ _p = 100 fs (see Section 2 of the Supplementary Material for more details). For ρ ≲ 20 and Q _1L ≲ 30, τ ^opt is positive, meaning that the switch should occur before the pulse enters the cavity. Moreover, for small Q _1L , τ ^opt strongly depends on ρ, while for large Q _1L , τ ^opt becomes almost independent from ρ (see the inset in Figure 2b). We note that as Q _1L increases, τ ^opt becomes negative for all values of ρ, meaning that the switching should occur once the pulse is already inside the cavity. The inset in Figure 2b shows that, for a fixed Q _1L , τ is critical only for small values of ρ. This is of particular interest since, in practice, small ρ are easier to induce and different ways to dynamically manipulate a cavity Q-factor have already been proposed [15, 16, 35, 44]. However, we stress that our model can be applied to a wider class of doubly resonant cavities beyond metasurfaces. Therefore, the results shown in Figure 2b allow identifying the relevant timescale regarding the tuning of the delay time τ.

In the following, we focus only on the dependence of SH conversion efficiency upon Q _1L and Q _1H , since a more detailed analysis of the exact dependence of τ ^opt on the other relevant parameters is beyond the scope of the present work. We calculate the optimum values of Q _1L and Q _1H , denoted as Q 1 L opt and Q 1 H opt , which maximize η _SH , by computing:

In Figure 3a, we report η _SH as a function of Q _1L for different ρ values. The black curve corresponds to the static case (Q ₁ = Q _1L = Q _1H → ρ = 1) and the purple triangle highlights the maximum SH conversion efficiency η S H max for the static case. We note that, at constant Q _1L , η _SH increases with increasing ρ. Indeed, an increasing Q _1H value corresponds to a decrease of the radiative losses of the FF mode; thus, the energy transferred from the FF to the SH mode (through the internal coupling term β ₂) increases. The vertical arrow highlights what happens in a Q-boosted cavity, i.e., fixed τ _p and ω ₁, for Q _1L = 55 (reference case in Figure 2a). A suitable choice of Q _1L and Q _1H values allows to enhance η _SH (blue circle) compared to the static case (violet square). The maximum conversion efficiency is attained for a Q _1L which is different from the optimal Q ₁ in the static case. We also note that Q 1 L opt decreases as ρ increases.

Figure 3:

Optimal configurations for efficiency enhancement. (a) Second harmonic conversion efficiency (η _SH ) at the optimum delay time (τ ^opt) as a function of the initial quality-factor (Q _1L ) for different values of ρ = Q _1H /Q _1L , where Q _1H is the final Q-factor: 1 (black curve, static case), 10 (gray curve), and 20 (yellow curve). The black vertical arrow highlights the η _SH enhancement for Q _1L = 55 (case displayed in Figure 2a). (b) Combinations of Q _1L and ρ at τ ^opt leading to maximum η _SH , for different impinging pulse duration (τ _p ). The value of η _SH is indicated by the colorscale. The inset reports the Q _1L corresponding to maximum η _SH as a function of τ _p . The black line corresponds to the bandwidth-matching condition ω ₁/(Q _1L ) = 1/τ _p . (c) η _SH as a function of ρ for different τ _p . For each ρ ^opt, it is possible to extract the corresponding Q 1 L opt from panel (b). In the panels, the pink triangles indicates the maximum conversion efficiency in the static case (ρ = 1).

In order to formulate useful guidelines for time-dependent doubly resonant cavities operation, we report in Figure 3b the trajectories (for various τ _p values), in the plane spanned by Q _1L and ρ, allowing a SH efficiency enhancement. Within a curve, each point represents the value of Q _1L which maximizes η _SH (i.e., Q 1 L opt ) at a given ρ and the colorscale highlights the corresponding SH efficiency value. At fixed τ _p , Q 1 L opt gradually shifts from the initial static case (ρ = 1, purple triangle) toward lower values and reaches, for increasing ρ, a minimum value Q 1 L min , which corresponds to η S H max (see Figure 3c). From Figure 3b inset, we note that Q 1 L min ∼ Q p = ω p τ p / ( 4 ln ⁡ 2 ) , where Q _p is an effective Q-factor given by the parameters describing the control pulse; therefore, Q 1 L min is close, but not equal, to the bandwidth-matching condition.

In order to qualitatively explain the relation between Q 1 L min and Q _p , we have to briefly discuss the properties of the spectral profile of the entities involved (see Section 3 of the Supplementary Material for more details). In the frequency domain, the spectral lineshape of the excited-wave amplitude within a resonant lossy system (at FF mode) is given by a Lorentzian function [25]

where Γ represents the spectral linewidth (FWHM). On the other hand, given the choice of the temporal profile of the control pulse in Equation (2), its spectral shape is given by a Gaussian function:

The SH conversion process involves an effective energy transfer from the input pulse s 1 + to the FF mode via the external coupling coefficient and then a transfer to the SH mode through the internal coupling term β. To do so efficiently, it is required to suitably match the spectral energy, within a certain spectral interval (delimited by ω _A and ω _B ), of the input pulse and the FF mode. The optimal matching condition between L F F and G , centered at ω ₁, is given by:

with ω _A,B = ω ₁ ∓ Δω _p , which leads to Γ ̃ / Δ ω p ≃ 0.4 < 1 (see Section 3 of the Supplementary Material). Thus, the matched cavity bandwidth Γ ̃ satisfies the relation: Γ ̃ < Δ ω p .

From Figure 3b, we note that the ρ value at which Q 1 L min occurs decreases with increasing τ _p , suggesting that, for long τ _p values, ρ ≃ 1. This is consistent with the results obtained for the monochromatic CW case [40] (corresponding to τ _p → ∞), for which a complete SH conversion efficiency is observed in the steady state regime of constant Q-factor (ρ = 1).

As noted in Figure 3c, η _SH decreases for ρ > ρ ^opt. This means that, after reaching ρ ^opt, a further increase of the cavity acceptance bandwidth is of little use, since the spectral weight of the frequencies far from ω ₁ is negligible. Ideally, despite the pump depletion term (β ₁) is included, for an infinite value of Q _1H and very long time, all the radiation stored in a ₁ can only escape through mode a ₂. This is because increasing Q _1H corresponds to closing the only radiation channel of mode a ₁, thus the energy can only escape the cavity through frequency conversion to ω ₂, since the radiation channel of mode a ₂ remains open. As expected from energy conservation η _SH < 1 and significantly increases for shorter pulses. It is clear that ρ ^opt decreases as τ _p becomes longer.

2.3 Limitations and implementation strategies

The modulation of the external coupling parameter Q ₁ may have consequences from a spectral point of view, thus potentially limiting the validity of the model. Indeed, an abrupt amplitude modulation of the Q-factor might: (i) introduce additional components in the spectrum of a ₁ (a ₂) located far from the central frequency ω ₁ (ω ₂) in the neighborhood of ω ₂ (ω ₁) or (ii) induce a significant frequency shift of the modes. Regarding the former aspect, the most delicate point is related to the switching time parameter σ, i.e., the speed at which the variation of Q ₁ occurs, compared to the optical cycle of the input pulse 2π/ω ₁. In our work, σ = 50 fs and τ _p = 100 fs. This means that, given the Q-factor dynamics in Equation (3), the time interval required to go from 10 % to 90 % of the entire variation is ∼ 70 fs, which is nearly 10 times larger than 2π/ω ₁ ∼ 7.9 fs. Therefore, although σ and τ _p are comparable, in the time interval 2π/ω ₁, the pulse intensity and the Q-factor can be assumed as constant. Regarding the latter aspect, to limit the impact of frequency shift in Q-boosting, specific conditions in refractive index and loss changes can be adopted in practical realizations [32], as demonstrated by FDTD simulations in Ref. [35]. The validity of our approach is further confirmed by continuous wavelet analysis of the modes dynamics [27], which reveal that the modes have a constant central frequency (see Section S5 of Supplementary Material).

For the effective implementation of this scheme in the case of broadband pulses, a large variation between the initial and final values of the cavity Q-factor is required to obtain a high SH conversion efficiency, as previously discussed. The practical realization of this can be achieved by the suitable design of a cavity endowed by a material-dependent sharp resonance, e.g., a quasi-BIC, at fundamental frequency. Thanks to a control light pulse, a variation of the material dielectric function [45, 46] can be induced, which in turn causes a large modulation of the Q-factor amplitude at FF with small frequency shift. A model including large frequency shift of the fundamental and/or SH modes is beyond the scope of this work. In order to avoid two-photon absorption and free carrier generation effects, an external control pulse with energy smaller than half of the bandgap may be used as in LiNbO₃ and other large bandgap materials with non-zero χ ⁽²⁾ [47]. Further theoretical development may take into account the role of non-radiative losses which may arise from free-carrier excitation or two-photon absorption. Other suitable platforms include dielectric slabs featuring a BIC mode which can be coupled to external radiation by means of a transient grating [48]. In such platforms the Q-factor can be dramatically increased in a short time by removing the transient grating, thus reducing any possible non-radiative loss and at the same time achieving a long-lived high-Q resonant condition.